We investigate the effect of loading rate on drainage in molecularly thin films of a simple fluid made of quasi-spherical molecules (octamethylcyclotetrasiloxane, OMCTS). We find that (i) rapidly confined OMCTS retains its tendency to organize into layers parallel to the confining surfaces, and (ii) flow resistance in such layered films can be described by bulklike viscous forces if one accounts for the existence of one monolayer immobilized on each surfaces. The latter result is fully consistent with the recent work of Becker and Mugele, who reached a similar conclusion by analyzing the dynamics of squeeze-out fronts in OMCTS [T. Becker and F. Mugele, Phys. Rev. Lett. {\bf 91} 166104(2003)]. Furthermore, we show that the confinement rate controls the nature of the thinning transitions: layer-by-layer expulsion of molecules in metastable, slowly confined films proceeds by a nucleation/growth mechanism, whereas deeply and rapidly quenched films are unstable and undergo thinning transitions akin to spinodal decomposition.